Question

Elyse and Jamir each make lemonade. Elyse mixes 1 2/3 cups of water with 3/4 cups of lemonade concentrate. Jamir mixes 1 1/3 cups if eayer witb 4/5 cups of lemonade concentrate. Which lemonade has a more lemony taste? Start by finding the unit rate for Elyse's recipe. You need to find the number of cups of lemonade concentrate per 1 cup of water. What can you divide 1 2/3 cups of water by to find 1 cup of water?

Answer 1

To find the unit rate for Elyse's recipe and determine the number of cups of lemonade concentrate per 1 cup of water, we can start by converting Elyse's mixture into an easier format.

Elyse's mixture consists of:
- Water: \( 1 \frac{2}{3} \) cups
- Lemonade concentrate: \( \frac{3}{4} \) cups

First, we need to convert \( 1 \frac{2}{3} \) cups of water into an improper fraction:

\[
1 \frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{5}{3} \text{ cups of water}
\]

Next, we want to find out how many cups of lemonade concentrate there are per 1 cup of water. We can do this by dividing the amount of lemonade concentrate by the amount of water and then scaling it to 1 cup of water.

To find the unit rate, we take the number of cups of lemonade concentrate and divide it by \( \frac{5}{3} \) (the amount of water), and then we adjust this so that we have it per 1 cup of water.

To do this, we use the formula:

\[
\text{Unit rate} = \frac{\text{Cups of lemonade concentrate}}{\text{Cups of water}} = \frac{\frac{3}{4}}{\frac{5}{3}}
\]

Dividing fractions involves multiplying by the reciprocal:

\[
= \frac{3}{4} \times \frac{3}{5} = \frac{9}{20}
\]

Thus, Elyse's lemonade has a concentration of \( \frac{9}{20} \) cups of lemonade concentrate per 1 cup of water.

To compare with Jamir's lemonade, we would perform a similar calculation for Jamir's recipe:

Jamir's mixture consists of:
- Water: \( 1 \frac{1}{3} \) cups
- Lemonade concentrate: \( \frac{4}{5} \) cups

Convert \( 1 \frac{1}{3} \) cups of water into an improper fraction:

\[
1 \frac{1}{3} = \frac{3 \times 1 + 1}{3} = \frac{4}{3} \text{ cups of water}
\]

Now, calculate the unit rate for Jamir:

\[
\text{Unit rate} = \frac{\frac{4}{5}}{\frac{4}{3}} = \frac{4}{5} \times \frac{3}{4} = \frac{3}{5}
\]

So Jamir's lemonade has a concentration of \( \frac{3}{5} \) cups of lemonade concentrate per 1 cup of water.

**Comparison:**
- Elyse's lemonade: \( \frac{9}{20} \)
- Jamir's lemonade: \( \frac{3}{5} = \frac{12}{20} \) when converted for comparison.

Since \( \frac{12}{20} > \frac{9}{20} \), Jamir’s lemonade is more lemony than Elyse's lemonade.